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Theorem fusgrvtxfi 29336
Description: A finite simple graph has a finite set of vertices. (Contributed by AV, 16-Dec-2020.)
Hypothesis
Ref Expression
isfusgr.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
fusgrvtxfi (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin)

Proof of Theorem fusgrvtxfi
StepHypRef Expression
1 isfusgr.v . . 3 𝑉 = (Vtx‘𝐺)
21isfusgr 29335 . 2 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
32simprbi 496 1 (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  cfv 6561  Fincfn 8985  Vtxcvtx 29013  USGraphcusgr 29166  FinUSGraphcfusgr 29333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-fusgr 29334
This theorem is referenced by:  fusgrfupgrfs  29348  nbfusgrlevtxm1  29394  nbfusgrlevtxm2  29395  nbusgrvtxm1  29396  uvtxnm1nbgr  29421  cusgrm1rusgr  29600  wlksnfi  29927  fusgrhashclwwlkn  30098  clwwlkndivn  30099  fusgreghash2wsp  30357  numclwwlk3lem2  30403  numclwwlk4  30405
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